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Re: circumference of an ellipse
*To*: mathgroup at smc.vnet.net
*Subject*: [mg19342] Re: circumference of an ellipse
*From*: "Alan W.Hopper" <awhopper at hermes.net.au>
*Date*: Fri, 20 Aug 1999 23:09:28 -0400
*Sender*: owner-wri-mathgroup at wolfram.com
Marcel,
Here is another approach to approximating the ellipse circumference
or perimeter, utilising a 'Complete Elliptic Integral of the 2 nd Kind'
which is incorporated in the Mathematica 3 function - EllipticE.
In[1]:= 4*a*Integrate[Sqrt[1 - e^2*Sin[Theta]^2],{Theta, 0, Pi/2}]
Out[2]= 4 a EllipticE[e^2]
In[3]:= a = 1; e = 0; 4 a EllipticE[e^2]
Out[4]= 2 Pi
Where, (e) Eccentricity = Sqrt[1-(b/a)^2]
In[5]:= ellipsePerimeter[a_,b_,n_]:= Module[{e = Sqrt[1-(b/a)^2]},
N[4 a EllipticE[e^2], n]]
In[6]:= ellipsePerimeter[2,1,10]
Out[7]= 9.688448221
In[8]:= ellipsePerimeter[100, 20, 30]
Out[9]= 420.20089079378001889398329177
It is interesting that in contrast to the above that there is a
very simple exact formula for the ellipse area ; A = Pi a b .
Alan W. Hopper
awhopper at hermes.net.au
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